An additive problem with an increasing number of summands (Q1425535)

This note continues investigations started in [\textit{R. N. Boyarinov} and \textit{V. N. Chubarikov} [Dokl. Math. 64, No. 1, 3--5 (2001); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 379, No. 1, 9--11 (2001; Zbl 1090.11502)]. The author studies the problem of the number of representations of a positive integer \(N\) in the form \(N = f(x_1) + f(x_2) +... +f(x_n)\), where \(1\leq x_1,x_2,...,x_n\leq Q\), \(x_1,x_2,...,x_n\) are positive integers, \(f(x)\) is a monotone sequence of positive integers, \(f(1) =1\), and \(n\) and \(Q\) increasing.